Theories of Hydrological Connectivity. Mike Kirkby Geography, U. Leeds

Theories of Hydrological Connectivity Mike Kirkby Geography, U. Leeds

Summary Structural and Functional connectivity Definitions etc Structural connectivity Connectivity for continuous and discrete systems Conductance and Percolation theory Importance of thresholds Humid and semi-arid systems Functional connectivity Progressive changes in a percolation network Limits of applicability to a hillslope/channel system Transitions between erosion and sedimentation in time and space Development of subsurface macropore networks Implications for morphology of pipe etc networks Periodic forms 2

Relationships between approaches investigating hydrological connectivity (from Bracken et al, 2013) 3

Structural and Functional connectivity Structural connectivity Refers to the spatial distribution landscape elements at present, and how they influence water transfer and flow paths May be static, dynamic or hysteretic Time invariant Dependent on system state (& history) Dependent on rate or direction of change of state Equilibrium commonly an implicit assumption Functional/ Progressive connectivity Refers to changes in the structural connectivity over time, most interestingly as a response to the interaction between the current structure of spatial elements and water/sediment flows driven by them. May involve positive or negative feedbacks associated with equilibrium tendencies or emergent properties. 4

Spatial contexts for structural hydrological connectivity Matrix Macropore/ Channel Network Dual porosity (i.e both active) Static response usually associated with permanently saturated conditions. Dynamic responses for free surface flow Surface water and some pipe/ macropore flow Hysteretic responses for flow in an unsaturated medium Scales of connectivity Micro (mm) Darcy/ Richards equations Hillslope (10 m) Overland and subsurface flow (TOPMODEL, LISEM) Catchment (100 km) Unit hydrograph etc 5

Types of connectivity metric for hydrological structure Static index (i.e. a single magic number) Nominal (i.e. On/Off) --? with a threshold Degree/ Probability of connection Connectivity function Index = Function (System state) e.g. Runoff coefficient = fn(% of surface wetted) Breakthough behaviour as one type of function Indices/ Functions may refer to Connection of one single point to another Many points or an area to a single outflow point or a stream bank Close relationship with concept of runoff coefficient 6

Conductance theory (Mcrae et al, 2008) Landscapes are represented as conductive surfaces, with low resistances (light grey in A) for habitats most permeable to movement, and high resistances (dark) barriers. B shows the lowest resistance pathways for faunal movement from bottom left to top right 7

Example contexts for applying conductance theory Ecology Faunal movement Migration of plant or animal species Under climate change scenarios Invasive species Fire susceptibility and spread Spread of HIV in Africa Conductance function based on access between areas. Graph theory as a subset of surface conductance. 8

Applicability of conductance theory in hydrology Simplest (static) case for a permanently saturated aquifer Hydraulic conductivity in unsaturated soil Varies with bulk moisture content Shows hysteresis (reflects wet/drying history) Near saturation, there is flow in macropores, which become increasingly dominant Flow mainly within conductive layers Access points vital for flow between layers e.g. cracks in hardpan layers or puncture points in surface crust 9

Percolation theory describes the behaviour of networks connecting adjacent sites or bonds in a random graph. The grid can be divided into a series of separate areas, each internally connected. sites or bonds. Here the white sites are occupied, and the ringed sites form a connected cluster. (Connections may be rectangular D4, as here, D6 or D8). The proportion of white is the level of occupancy. It is easy to conceptualise occupied sites as patches of surface water, each with 4 possible 10 neighbours

Connections with different occupancy levels 40% occupancy: No though connection 70% occupancy: Through connection 11

Probability of connection has sharper threshold as size of grid, L, increases For this square grid, the percolation threshold (PT) occurs at the occupancy level of 0.59 12

For a given occupancy, P(through connection) increases and Percolation Threshold decreases with number of neighbours and with dimensionality Triangular pattern. Each site has connections to 6 neighbouring sites. PT = 0.50 Square pattern. Each bond has connections to 6 neighbouring bonds. PT = 0.50 Simple 3-D cubic pattern. Each site has connections to 6 neighbouring sites. PT = 0.31 13

Hydrological implications Occupancy is related, fairly intuitively, to the amount of water on a surface As hydrologists, we are interested in 1. The runoff coefficient or partial contributing area At its simplest, this is the proportion of the wetted area that is linked to the outflow margin 2. Whether there is a connection across the domain of interest 14

These graphs show the mean of five replications in a square 50x50 grid. The upper graph shows the probability of a through connection from top to bottom of the grid. The simulation confirms the sharp PT of 0.59. The lower graph shows the proportion of the occupied (?wetted) cells that are connected to the lower boundary. Although less sharp, the PT is still very evident 15

Relative Surface Connection Function (RSCF) (Antoine, Javaux, Bielders Adv Water Res 2009) Fraction of wetted surface connected to outlet These curves show a similar general form, generated in a more directly hydrological context. 16

Simulation of infiltration excess runoff in a square grid Possible destination map of overland flow from starting cell during one time step Accumulation of runoff through the grid with potential for re-infiltration Gradients driving any overland flow successively to all or some lower neighbours during each time step Multiple flow paths with power law weighting proportional to gradient ^n (for n=1-3) Soil infiltration parameters randomly defined for each cell in the grid and then fixed over time. OF velocity ~ depth 1/2 x local gradient. 17

DEM contours Catchment areas (n=3) 18

30 mm storm over 30 minutes Variable velocity of overland flow Total runoff = 7% Simulation below with 500 OF paths Mean overland flow velocity (cells per time interval) 19

Y Axis Title 22

Simulated % runoff for 30 minute storms on 2% to 30% gradients, showing marked threshold behaviour R0=80mm Runoff Coefficient = 1/[1+(R 0 /R) 3 ] Runoff threshold, R 0, increases with slope length & storm duration; and decreases with slope gradient. 60

Take home message for discrete ephemeral networks In a non-evolving network, connected flow is a threshold phenomenon Threshold can be linked to event size Can be expressed as a non-linear rainfallrunoff relationship Or as a breakthrough curve Breakthrough is less abrupt for connected area than for presence/absence of a through connection. Thresholding is more abrupt for larger areas Example of semi-arid floods 61

Contrast in nature of connectivity between humid and semi-arid sites SEMI-ARID SITE HUMID SITE Rf E-T A Rf E-T B SEMI-ARID Common Rf drivers Local patchiness in E-T, constrained by available moisture Patchy infiltration capacity Connectivity through: Surface Local through Episodic OF Subsurface Lateral Competition for E-T by roots Normally zero SS runoff Connectivity generally decreases with slope length HUMID Common Rf drivers E-T variation driven by sub-surface moisture differences Patchy infiltration capacity Connectivity through: Surface Episodic OF near slope base Subsurface Lateral flow driving other links Runoff normally more or less spatially uniform Connectivity generally increases with 62 slope length

Dynamic expansion of connected storm runoff areas in SE Spain (from Cammeraat, 2002*) Semi-arid environment: Subsurface flow and connectivity is negligible. First-order assumption is that subsurface exchanges are entirely vertical. Overland flow creates transitory (and partial) connections that re-set the system to a common state. *A Review Of Two Strongly Contrasting Geomorphological Systems Within The Context Of Scale. ESPL, 27, 1201 1222 63

Example of partially connected overland flow, Wharfedale, UK (from Lane et al, 2004): HP 18, 191-201 Humid environment: Subsurface connectivity all the time, driven by lateral flows. First Order approximation is that subsurface runoff is uniform over space. Surface connectivity only through connected saturated zones. 64

Structural/Progressive connectivity Connections established in a flood convey sediment that modifies the quality of the connectivity, generally increasing the probability of connection or runoff coefficient and/or lowers the threshold. Is this a one-way process, or does it lead to an equilibrium, perhaps exhibiting the emergent property of a stable channel/pipe network 65

Experiment in progressive connectivity After each run of a random network, with a random total occupancy (0-100%) the probability of occupancy for each cell connected to the outlet is increased for future runs as follows: Threshold (initially 100%) reduced by 5-10% Probability of occupancy =0.5(2- threshold) Sites are occupied at random until total occupancy target has been reached. 66

Threshold patterns in a 20x20 grid after 100 runs # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # Cells with threshold in lowest quartile. (Median=18: Lower quartile=15 Range = 5 to 32) Relationship between Runoff (% of occupied area draining to outlet) and Rainfall (% occupancy) Later runs do not significantly greater runoff as a result of lower thresholds near outlet. 67

Connectivity-relevant transitions in the hillslope/channel system In any event, there are two key thresholds for sediment behaviour along a flow line from divide to the sea. 1. The point at which the flow tractive stress is first able to entrain the available sediment, allowing formation of an eroded channel 2. The point at which the stream in no longer to carry all of the sediment eroded from upstream, and begins to deposit some of it. In larger (areally uniform) storm events, these points generally both migrate upstream. Over the frequency distribution of storms, these transition points move around some average position, and also leave an imprint of the last major event. The upper transition point is the stream head and the lower point is the headward tip of the flood plain. 68

Connectivity above and below the stream head Above the stream head, diffusive sediment movement (e.g. by rain splash and soil creep) tends to smooth out any irregularities, so that connectivity is not progressive. Once a channel is formed, good connectivity is etched into the surface For a semi-arid area, slope lengths are shortened and connected runoff increased. For a humid area, runoff is controlled by subsurface flow, so that the effect is much weaker. 69

Connectivity and the flood plain Flood plain sediment absorbs some runoff, particularly on rising stage of a flood, and allows spreading and slowing of overbank flows. In ephemeral streams, flood plain sediment raises the breakthrough threshold, reducing connectivity. In humid streams, connectivity is reflected by diffusive and advective flood routing parameters. Flood plain sediment and space for overbank flow increases diffusion and reduces advective velocity, also decreasing connectivity. 70

Thought experiment for establishing a new subsurface macropore connection Surface connection to supply of ponded water Length, L Cylindrical tube of radius r Hydraulic gradient down the slope, G Assume that enlargement of the tube is a function of the amount removed (as sediment or in solution), and this rate of removal is proportional to Q Discharge down the tube, Q 71

Surface connection to supply of ponded water Length, L Cylindrical tube of radius r Hydraulic gradient down the slope, G The Math For laminar flow, Q= G r 4 /(8 ) Total loss from pipe walls = Q = G r 4 /(8 ) This loss is spread over an area of 2 r L, so that the rate of pipe enlargement, dr/dt = Q /(2 r L) = Gr 3 /(16 L) Solving this equation for a pipe of initial radius r 0 1/r 2 =1/r 0 2 Gt/(8 L) where t is the time during which the pipe inlet is in ponded state. Discharge down the tube, Q Conclusions: Connectivity is established rapidly after initial very slow growth. As pipe grows wider, infinite growth is prevented because pipe can eventually carry more water than is available. Time to connection is least when L small and G, r 0 large. Time will be shorter where water is most frequently available at source. Pipes will preferentially join to existing pipes (small L), creating dendritic net. 72

Periodic forms of connectivity Water driven Tiger bush etc. striped vegetation Sediment driven by wind or water Ripples and dunes in sand Gravel bars and rocky deserts Patterned ground in cold climates Complex dynamics often generating Transverse forms under low movement rates Longitudinal forms under high movement rates 73

Generalised math for tiger bush Assumptions Continuity of mass: Discharge: Hydrology and Veg n water use q = q 0 h = a - q t x s h x a = R e. h where h= water depth q= discharge per unit width t =elapsed time x =distance downslope a= net gain/loss q 0 = discharge parameter s =ground slope R =runoff rate e= rate of water use by veg n Deduction h = R e sin x e q 0 1 2 + 1 This is periodic with wavelength, = 2 (q 0 /e) 1/2 74

Key conclusions 1. Hydrological connectivity is commonly associated with rather sharp thresholds. 2. Establishment of connection is better described as a function of the driver (e.g. storm volume) than by a single index. 75